About the image of strongly generalized derivations of order n

Let A and B be two algebras. A linear mapping 𝜟:A → B is called a strongly generalizedderivation of order n, if there exist the families {𝐸_𝑘: 𝐴 → 𝐵}𝑘 = 1𝑛, {𝐹_𝑘: 𝐴 → 𝐵}𝑘 = 1𝑛, {𝐺_𝑘: 𝐴 → 𝐵}𝑘 = 1𝑛 𝑎𝑛𝑑 {𝐻_𝑘: 𝐴 → 𝐵}𝑘 = 1𝑛 of linear mappings which satisfy 𝛥(𝑎𝑏) = Σ [𝐸_𝑘(𝑎) 𝐹_𝑘(𝑏) + 𝐺_𝑘(𝑎)𝐻_𝑘(𝑏)] 𝑛𝑘 =1 for all 𝑎, 𝑏 ⋲ 𝐴.The main purpose of this paper is to study the image of such derivations. Our main result onthe image of strongly generalized derivations of order one reads as follows: Let A be a unital,commutative Banach algebra and let 𝛥: 𝐴 → 𝐴 be a continuous strongly generalizedderivation of order one; that is, there exist the linear mappings 𝐸, 𝐹, 𝐺, 𝐻: 𝐴 → 𝐴 satisfying𝐷(𝑎𝑏) = 𝐸(𝑎) 𝐹(𝑏) + 𝐺(𝑎) 𝐻(𝑏) for all 𝑎, 𝑏 ⋲ 𝐴. Let 𝐸, 𝐹, 𝐺 and 𝐻 be continuous linearmappings. We prove that, under certain conditions, 𝐻 (𝐴), 𝐸(𝐴) 𝑎𝑛𝑑 𝛥(𝐴) are contained inthe Jacobson radical of A. This result generalizes Singer-Wermer theorem about the image ofcontinuous derivations on commutative Banach algebras.